hello to all

I have some issue of understanding this criteria... it says:

my question is why it is "uniformly" convergent ? I just can't see the fact why we have here uniformly convergent series...Let

$\displaystyle \displaystyle (a_n)_{n\in \mathbb{N}$

be positive numerical sequence for which is worth that for almost every

$\displaystyle \displaystyle n \in \mathbb{N}$

and every

$\displaystyle \displaystyle x\in A$

is satisfied that

$\displaystyle \displaystyle |f_n (x)|\le a_n$

then if series:

$\displaystyle \displaystyle \sum_{n=1} ^{+\infty} a_n $

is convergent then series (of functions) :

$\displaystyle \displaystyle \sum_{n=1} ^{+\infty} f_n(x) $

is uniformly convergent on set A.

Thanks for any help